Page:EB1911 - Volume 09.djvu/161

13. Besides enabling us to express the extension in aniy direction and the changes of relative direction of any filaments of the body, the components of strain also express the changes of size of volumes and areas. In particular, the “cubical dilatation,” that is to say, the increase of volume per unit of volume, is expressed by the quantity $e_{xx}+e_{yy}+e_{zz}$ or ${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}\cdot$ When this quantity is negative there is “compression.”

Fig. 3

14. It is important to distinguish between two types of strain: the “rotational” type and the “irrotational” type. The distinction is illustrated in fig. 3, where the figure ${\text{A}}''{\text{B}}''{\text{C}}''{\text{D}}''$ is obtained from the figure ${\text{ABCD}}$ by contraction parallel to ${\text{AC}}$ and extension parallel to ${\text{BD}},$ and the figure ${\text{A}}'{\text{B}}'{\text{C}}'{\text{D}}'$ can be obtained from ${\text{ABCD}}$ by the same contraction and extension followed by a rotation through the angle ${\text{A}}''{\text{OA}}'.$ In strains of the irrotational type there are at any point three filaments at right angles to each other, which are such that the particles which lie in them before strain continue to lie in them after strain. A small spherical element of the body with its centre at the point becomes a small ellipsoid with its axes in the directions of these three filaments. In the case illustrated in the figure, the lines of the filaments in question, when the figure ${\text{ABCD}}$ is strained into the figure ${\text{A}}''{\text{B}}''{\text{C}}''{\text{D}}'',$ are ${\text{OA, OB}}$ and a line through ${\text{O}}$ at right angles to their plane. In strains of the rotational type, on the other hand, the single existing set of three filaments (issuing from a point) which cut each other at right angles both before and after strain do not retain their directions after strain, though one of them may do so in certain cases. In the figure, the lines of the filaments in question, when the figure ${\text{ABCD}}$ is strained into ${\text{A}}'{\text{B}}'{\text{C}}'{\text{D}}',$ are ${\text{OA, OB}}$ and a line at right angles to their plane before strain, and after strain they are ${\text{OA}}',{\text{OB}}'$ and the same third line. A rotational strain can always be analysed into an irrotational strain (or “pure” strain) followed by a rotation.

Analytically, a strain is irrotational if the three quantities

${\frac {\partial w}{\partial y}}-{\frac {\partial v}{\partial z}},$⁠${\frac {\partial u}{\partial z}}-{\frac {\partial w}{\partial x}},$⁠${\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}$

vanish, rotational if any one of them is different from zero. The halves of these three quantities are the components of a vector quantity called the “rotation.”

15. Whether the strain is rotational or not, there is always one set of three linear elements issuing from any point which cut each other at right angles both before and after strain, If these directions are chosen as axes of ${x,y,z,}$ the shearing strains ${e_{yz},e_{zx},e_{xy}}$ vanish at this point. These directions are called the “principal axes of strain,” and the extensions in the directions of these axes the “principal extensions.”

16. It is very important to observe that the relations between components of strain and components of displacement imply relations between the components of strain themselves. If by any process of reasoning we arrive at the conclusion that the state of strain in a body is such and such a state, we have a test of the possibility or impossibility of our conclusion. The test is that, if the state of strain is a possible one, then there must be a displacement which can be associated with it in accordance with the equations (1) of § 11.

We may eliminate ${u,v,w}$ from these equations. When this is done we find that the quantities ${e_{xx},\dots e_{yz}}$ are connected by the two sets of equations

⁠ ${\frac {\partial ^{2}e_{yy}}{\partial z^{2}}}+{\frac {\partial ^{2}e_{zz}}{\partial y^{2}}}+{\frac {\partial ^{2}e_{yz}}{\partial y\partial z}}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ ⁠. ⁠ . ⁠ . ⁠ (1)
⁠ ${\frac {\partial ^{2}e_{zz}}{\partial x^{2}}}+{\frac {\partial ^{2}e_{xx}}{\partial z^{2}}}+{\frac {\partial ^{2}e_{zx}}{\partial z\partial x}}$
⁠ ${\frac {\partial ^{2}e_{xx}}{\partial y^{2}}}+{\frac {\partial ^{2}e_{yy}}{\partial x^{2}}}+{\frac {\partial ^{2}e_{xy}}{\partial x\partial y}}$

and

$2{\frac {\partial ^{2}e_{xx}}{\partial y\partial z}}+{\frac {\partial }{\partial x}}\left(-{\frac {\partial e_{yz}}{\partial x}}+{\frac {\partial e_{zx}}{\partial y}}+{\frac {\partial e_{xy}}{\partial z}}\right)$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ ⁠ . ⁠ . ⁠ (2)
$2{\frac {\partial ^{2}e_{yy}}{\partial z\partial x}}+{\frac {\partial }{\partial y}}\left({\frac {\partial e_{yz}}{\partial x}}-{\frac {\partial e_{zx}}{\partial y}}+{\frac {\partial e_{xy}}{\partial z}}\right)$
$2{\frac {\partial ^{2}e_{xx}}{\partial y\partial z}}+{\frac {\partial }{\partial z}}\left({\frac {\partial e_{yz}}{\partial x}}+{\frac {\partial e_{zx}}{\partial y}}+{\frac {\partial e_{xy}}{\partial z}}\right)$

These equations are known as the conditions of compatibility of strain-components. The components of strain which specify any possible strain satisfy them. Quantities arrived at in any way, and intended to be components of strain, if they fail to satisfy these equations, are not the components of any possible strain; and the theory or speculation by which they are reached must be modified or abandoned.

When the components of strain have been found in accordance with these and other necessary equations, the displacement is to be found by solving the equations (1) of § 11, considered as differential equations to determine ${u,v,w}.$ The most general possible solution will differ from any other solution by terms which contain arbitrary constants, and these terms represent a possible displacement. This “complementary displacement” involves no strain, and would be a possible displacement of an ideal perfectly rigid body.

17. The relations which connect the strains with each other and with the displacement are geometrical relations resulting from the definitions of the quantities and not requiring any experimental verification. They do not admit of such verification, because the strain within a body cannot be measured. The quantities (belonging to the same category) which can be measured are displacements of points on the surface of a body. For example, on the surface of a bar subjected to tension we may make two fine transverse scratches, and measure the distance between them before and after the bar is stretched. For such measurements very refined instruments are required. Instruments for this purpose are called barbarously “extensometers,” and many different kinds have been devised. From measurements of displacement by an extensometer we may deduce the average extension of a filament of the bar terminated by the two scratches. In general, when we attempt to measure a strain, We really measure some displacements, and deduce the values, not of the strain at a point, but of the average extensions of some particular linear filaments of a body containing the point; and these filaments are, from the nature of the case, nearly always superficial filaments.

18. In the case of transparent materials such as glass there is available a method of studying experimentally the state of strain within a body. This method is founded upon the result that a piece of glass when strained becomes doubly refracting, with its optical principal axes at any point in the directions of the principal axes of strain (§ 15) at the point. When the piece has two parallel plane faces, and two of the principal axes of strain at any point are parallel to these faces, polarized light transmitted through the piece in a direction normal to the faces can be used to determine the directions of the principal axes of the strain at any point. If the directions of these axes are known theoretically the comparison of the experimental and theoretical results yields a test of the theory.

19. Relations between Stresses and Strains.—The problem of the extension of a bar subjected to tension is the one which has been most studied experimentally, and as a result of this study it is found that for most materials, including all metals except cast metals, the measurable extension is proportional